The shape function method of nonlinear thermal stress of granite fracture tips in a high-temperature environment

Exposed rock masses in tunnel portals are susceptible to thermal deterioration in southern China, where temperatures are relatively high. The thermal stress field of rock masses is affected by fracture shape and distribution as fractures near the surface are channels for solar radiation energy to be converted into rock thermal energy. In this study, a function expression is developed for triangular heat sources of fractured rock masses in a tunnel portal in a high-temperature environment. By the function expression, the temperature field and thermal stress field are calculated, and the influence of fracture shape parameters and multi-fracture interaction is analyzed. The results are as follows: (1) the temperature field and thermal stress field of exposed rocks are redistributed by fractures. The internal temperature of the fractured rocks is higher than that of non-fractured rocks, and thermal stress near the fracture tip increases. (2) For triangular fractures of the same length, thermal stress increases as the apex angle increases. (3) When the spacing between parallel fractures or coplanar fractures is close, the superposition effect of thermal stress becomes significant. (4) In a high-temperature environment, temperature field and thermal stress field of a fractured rock are both nonlinear as temperature and thermal stress around fractures increase significantly. The results provide effective reference for stability evaluation of fractured rock masses in tunnel portals and offer theoretical foundation for thermal diseases analysis and protection measures of tunnel engineering in high-temperature environments of southern China.

The thermal stress field of fracture tip of a rock in a high-temperature environment The thermal effect of a fractured rock in a high-temperature environment In a high-temperature environment, the process of heat exchange between exposed rocks and the external environment is considered a comprehensive effect of solar radiation, radiation heat transfer and convection heat transfer on the surface of a semi-infinite solid.When there are surface fractures in a rock, the heat exchange process presents that energy of solar radiation and atmospheric radiation is absorbed by fractures and converted into thermal energy of rock, and convection heat transfer is unavailable as the air in fractures does not flow.
The heat exchange between a fractured rock and the external environment is shown in Fig. 2. Radiation energy is absorbed and reflected by fracture fronts and then decays.Absorption and reflection repeat several cycles in a fracture and then all radiation energy has been absorbed or emitted through the fracture, as fracture shapes are long and narrow, and fracture apertures are small.The solar radiation absorption coefficient is defined as the ratio of solar radiation energy absorbed by a rock to the total solar radiation energy reaching the rock.The absorption coefficient of rock to solar short-wave radiation and atmospheric long-wave radiation is 0.85 and

The temperature field of fracture tip of a rock
The thermal effect of exposed rocks in high-temperature environments is temperature variation caused by heat flux transmitted through medium at the boundary of half-space.According to the theory of heat conduction, rock temperature can be expressed as where T is temperature (°C), t is time (s), a = λ/ρc is thermal diffusivity of a rock (m 2 s −1 ), λ, ρ, c are thermal conductivity (W m −1 K 1 ), density (kg m −3 ) and heat capacity (J kg −1 K −1 ) of a rock, respectively.
As a result, the following formula can be obtained where ΔT is temperature difference (°C), T = T − T 0 , T 0 is initial rock temperature (°C).The boundary condition of heat exchange between a rock surface and an environment could be expressed as where q 1 = α ′ q s + εq r + q h , q s , q r , q h are heat flux density of solar radiation, radiation heat transfer and convec- tion heat transfer (W m −2 ), respectively 42 .α′ and ε are absorption coefficients of a rock to solar radiation and atmospheric radiation.
Effective radiation F = εq r is determined by utilizing the method of expanding temperature amplitude.Thus, air temperature function could be expressed as: where T M is mean daily temperature (°C), T A is daily temperature amplitude (°C), t 0 is time parameters of maximum and minimum temperature, C f is effective radiation influence coefficient 43 .
Analytical solution of temperature field of unsteady heat conduction in a semi-infinite solid under the second boundary condition could be obtained by 42 : Analytical solution under the third boundary condition could be obtained by: The heat exchange between a triangular fractured rock and the external environment 11,12 .
where h is convective heat transfer coefficient of a rock (W m −2 K −1 ).In a high-temperature environment, temperature field of a rock without fracture could be written as: When there are near surface fractures in a rock, the fracture fronts can be considered as finite line heat sources with uniform heat flux as fractures absorb radiation energy and convert it into heat energy (Fig. 3).The boundary condition is that the temperature of medium surface (the ground temperature) is uniform and equal to the temperature of medium at infinity (the initial rock temperature T 0 ).
On a fracture front: where q 2 = q s + q a , q a is heat flux density of atmospheric radiation (W m −2 ) 42 .The Green's Function Solution of temperature increment produced by a unit instantaneous point heat source (x 0 , y 0 , z 0 ) at t 0 in an infinite rock could be expressed as follow 44 : where ΔT 0 is temperature difference generated by a point heat source (°C), R is distance from the point heat Temperature increment produced by a point heat source (x 0 , y 0 , z 0 ) with intensity q 2 in time period t m−1 ∼ t m can be given by: The time of diurnal insolation is divided into M hours.If intensity of heat source remains constant in the mth hour with value of q 2 (t m ), temperature increment of a fractured rock during the insolation period can be given by: Temperature increment caused by a triangular fracture line heat source can be given by: where Γ is a triangular fracture line heat source, Ŵ: is apex angle of a triangular fracture (°), L is length of a fracture (m), H is length of a fracture opening (m), the position relationship is shown in Fig. 3. Using the mirror image method, a mirror heat source with an intensity of − q 2 (t m ) is set at a position symmetrical to the line heat source Temperature increment caused by superposition of all k heat sources (line heat sources and mirror heat sources) can be given by: In a high-temperature environment, temperature field of fracture tip of a rock can be obtained by: The thermal stress field of fracture tip of a rock Thermal stress occurs in an exposed rock due to the temperature distribution is not uniform and the thermal expansion and contraction of each point of a rock are inconsistent under influence of insolation in high-temperature environments.Temperature change caused by rock deformation is ignored and coupling term of heat conduction equation is not taken into account because the temperature change is slow.Equilibrium equation of a rock is where σ ij is thermal stress of a rock (MPa), i and j take x, y or z.
When temperature of a rock changes, the constitutive equation is where Boundary condition of a semi-infinite rock is In a high-temperature environment, stress field of an exposed non-fractured rock resulting from thermal effect can be expressed as 45 : When there are near surface fractures in a rock, fracture fronts are regarded as finite line heat sources with uniform heat flux.The Green's Function Solution of thermal stress field caused by a unit instantaneous point heat source (x 0 , y 0 , z 0 ) at t 0 in an infinite rock could be expressed as 45 : where σ 0 ij is thermal stress generated by a point heat source (MPa), i and j take x, y or z.Thermal stress field generated by a point heat source (x 0 , y 0 , z 0 ) with intensity q 2 in time period t m−1 ∼ t m can be given by: The time of diurnal insolation is divided into M hours.If heat source intensity remains constant in the mth hour with a value of q 2 (t m ), thermal stress field of a fractured rock during the insolation period could be expressed as: For a semi-infinite rock, boundary conditions can be satisfied by mirror image method except for shear stress on the boundary (σ xz and σ yz ).Thermal stress field produced by superposition of all k heat sources (line heat sources and mirror heat sources) could be expressed as: The shear stresses on the boundary (σ xz and σ yz ) caused by line heat sources and mirror heat sources are the same.Therefore, solutions of the following boundary value problem should be superimposed on the half space problem: where f x , f y are sum of the shear stresses on boundary caused by line and mirror heat sources (MPa).
Stress field in half space created by concentrated shear stress f x (ξ, η, 0), f y (ξ, η, 0) acting on the point (ξ, η, 0) of boundary could be calculated by: As a result of integration of the influence coefficients The boundary surface can be discretized into S rectangular elements.Assuming the shear stress at each element is uniformly distributed, the stress field in half space caused by shear stress on the boundary could be obtained by: where s is the sth rectangular element , f sx and f sy are values of f x and f y at center of the sth element, respectively, ξ s1 , ξ s2 are region restriction in x direction of the sth element, η s1 , η s2 are region restriction in y direction of the sth element.
Thermal stress field of a single fracture tip in a high-temperature environment could be obtained by In the case of N fractures in a rock, Eq. ( 28) represents the thermal stress field caused by the nth fracture.The thermal stress field of a multi-fractured rock could be obtained by A brittle fracture of a rock satisfies the theory of maximum tensile stress.Stress intensity factors of a surface fracture could be expressed as: where K I and K II are stress intensity factors of a fracture, κ is correction coefficient of stress intensity factor of a surface fracture.
Local coordinate system is established using a fracture tip point as origin.Vertical fracture direction is y′ direction, fracture extension direction is z′direction, and direction perpendicular to y′z′ plane is x′ direction.Coordinates can be converted as: Circumferential tensile stress near a fracture tip could be expressed as: where r, θ are polar diameter and polar angle in coordinate system x′oy′, x′ = rcosθ, y′ = rsinθ.
When maximum circumferential tensile stress exceeds tensile strength of a rock, i.e.σ θ max ≥ [σ m ] , a fracture will expand.

Influence of a fracture on rock temperature field
Fractures are channels through which solar radiation energy is converted into rock thermal energy, affecting rock temperature field.The influence of a fracture on temperature field in a high-temperature environment is studied by using granite in Guangzhou section of the Guangzhou-Shenzhen High-speed Railway as an example.Before sun rises, temperature field of a rock is approximately uniform and this temperature is regarded as the initial rock temperature.Parameters of an environment and granite properties 46 are shown in Fig. 4 and Table 1.The fracture shape is an isosceles triangle.The length of a fracture is 0.35 m and the length of a fracture opening is 0.14 m.

Analysis of temperature field of a fracture tip
In solar radiation period, the fractured rock temperatures at depths of 0.05 ~ 0.65 m from the surface (A1 ~ A7 in Fig. 5) are calculated and compared with non-fractured rock temperatures (Fig. 6).During 6:00 ~ 8:00, the difference in temperature between the fractured rock and the non-fractured rock at a given depth is insignificant (27)   Table 1.Parameters of an environment and granite properties 46 .because of weak radiation.From 9:00 to 18:00, rock temperature increases due to heat effect of solar radiation.The rate of temperature increase is consistent with the changes in solar radiation intensity and reaches its maximum at 13:00.Additionally, the temperature difference between the fractured rock and non-fractured rock increases.Temperature changes of the non-fractured rock are the result of combination of solar radiation, radiation heat transfer and convection heat transfer.When there is a fracture in the rock, convection heat transfer is impossible as the air in a fracture does not flow and the radiation energy cannot escape from the fracture.Therefore, temperature changes of a fractured rock are determined only by the intensity of solar radiation.The temperature of a fractured rock is higher than that of a non-fractured rock at the same depth, and it varies rapidly and greatly.For a fractured rock under solar radiation, the fracture tip has strong physical and geometric effect, which presents a strong ability to absorb energy and leads to heat and energy concentration.The temperature field of a fracture tip is nonlinear and the temperature of the fracture tip is high.Under the fracture tip, the temperature decreases sharply and eventually approaches the initial rock temperature along z axis with depth.In a high-temperature environment, the change in rock temperature field caused by a fracture is primarily confined to the areas around the fracture and has little effect on distant areas.

Analysis of fracture shape parameters
In a high-temperature environment, the distribution of rock temperature field is affected by the fracture shape.In order to analyze the influence of shape parameters, temperature fields of typical triangular (acute angle, right angle, obtuse angle) fractured rocks with the same a fracture length of 0.35 m are calculated at 13:00 (Fig. 7).The apex angles of triangular fractures are 60°, 90° and 120°, respectively.The rock temperature increases with the increase of apex angle of a triangular fracture and reaches its maximum when the apex angle is 120°.It is observed that the temperature of the triangular fractured rocks with a constant fracture length vary greatly when the apex angle is large in a high-temperature environment.

Analysis of thermal stress field of a fracture tip
In order to examine the impact of a fracture on rock thermal stress field in a high-temperature environment, thermal stress of a fractured rock at depths of 0.05 ~ 0.65 m from surface (A1 ~ A7 in Fig. 5) is calculated and compared with that of a non-fractured rock (Fig. 8).In the fractured rock and the non-fractured rock, the daily variation trends of thermal stress are all consistent with that of the solar radiation.When the solar radiation is weak between 6:00 and 8:00, the thermal stress of the fractured rock and the non-fractured rock are basically unchanged.From 9:00 to 13:00, when the solar radiation increases gradually, the thermal stress increases sharply at first and then tends to be gentle.At 13:00, the maximum of thermal stress occurs.From 14:00 to 18:00, when the solar radiation weakens, the thermal stress gradually decreases.
The thermal stress field of a rock is redistributed due to a fracture.Horizontal stresses σ x , σ y of a non-fractured rock concentrate at the surface layer and decrease rapidly to 0 as depth increases.Vertical stress σ z of a nonfractured rock is 0. Compared with a non-fractured rock, the stress concentration area in a fractured rock is formed near the fracture tip.The σ x first increases and then decreases with depth and the maximum value is 0.53 MPa at the depth of 0.25 m.The maximum value of fracture tip(A4) is 0.42 MPa.The σ y and σ z increase with depth, and the maximum values are 0.55 and 0.85 MPa at fracture tip, respectively.Under the fracture tip, the thermal stress decreases sharply with depth and finally approaches 0. Fractures provide channels for transfer and transformation of solar radiation energy to the interior of rocks, which is equivalent to adding linear heat sources in rocks.Therefore, fractures change the internal heat conduction law and redistribute the thermal stress field of rocks.Compared with a non-fractured rock, horizontal stress σ x , σ y of a fractured rock are larger and vertical stress σ z occurs in a fractured rock.Thus, thermal stress of a fractured rock at the depth of fracture tip increases compared with non-fractured rock in the same depth and thermal stress field near a fracture tip is nonlinear, which is easy to cause cracking and lead to rock instability.The change in rock thermal stress field caused by a fracture in a high-temperature environment is concentrated in areas around the fracture and has little influence on distant areas.
The circumferential tensile stress near a fracture tip is analyzed using relative critical size α = 2r c H .The results are equivalent when either critical size r c or relative critical size α approaches 0. The influence of α on maximum circumferential tensile stress σ θmax is demonstrated in Fig. 9. Since the relative critical size α is small, the maximum circumferential tensile stress σ θmax is great.In other words, the change rate and daily amplitude  www.nature.com/scientificreports/variation of σ θmax is great near a fracture tip.During the period of insolation, the gradient of σ θmax increases first and then decreases, reaching its maximum at 13:00.

Analysis of fracture shape parameters
In a high-temperature environment, thermal stress of a rock is affected by fracture shape.For analysis of the influence of fracture shape parameters on thermal stress field of a rock, the maximum circumferential tensile stresses σ θmax near tips of typical triangular (acute angle, right angle and obtuse angle) fractures with the same length of 0.35 m are calculated.The apex angles of triangular fractures are 60°, 90° and 120°, respectively.The changes in σ θmax of three typical triangular fractured rocks are analyzed at 13:00 when the thermal stress reaches its maximum (Fig. 10).For rocks with triangular fractures of the same length in a high-temperature environment, the σ θmax increases with the increase in apex angles of triangular fractures, reaching a maximum when the apex angle is 120°.The stress gradient is high as the apex angle of triangle is large.Additionally, σ θmax does not change greatly as α increases.

Analysis of fracture interaction
Numerous fractures are disorderly distributed in an engineering rock mass.Interaction between fractures is characterized by a superposition of stress fields of fracture tips resulting in fracture connection and rock instability.
Considering the effect of fracture number and distribution on thermal stress field, maximum circumferential tensile stress σ θmax near fracture tips are studied for a rock with double-fracture and triple-fracture in a hightemperature environment.

Parallel fractures interaction
The maximum circumferential tensile stress σ θmax near a tip of parallel fractures is calculated and the change in stress compared to a single fracture is analyzed at 13:00 with a length of fracture of 0.35 m and a length of fracture opening of 0.14 m (Fig. 11).Vertical spacing d is 0.02, 0.04, 0.06, 0.08, 0.10, 0.12 and 0.14 m, respectively.www.nature.com/scientificreports/U is set as percentage change in stress between multi-fracture compared to a single fracture.The thermal stress field of a fracture tip is superposed due to interaction of fractures.Fracture spacing affects the superposition.Maximum circumferential tensile stress σ θmax is analyzed by setting V as a ratio of a vertical fracture spacing d to a fracture length L. When the distance between parallel fractures is small (V ≤ 0.20 of double-fracture, V ≤ 0.37 of triple-fracture), the σ θmax near a fracture tip increases more than 20% and the interaction of fractures produces a significant increase in σ θmax .When there is a certain distance between parallel fractures (0.20 < V ≤ 0.60 of double-fracture, 0.37 < V ≤ 0.89 of triple-fracture), the fracture interaction is weak.When the distance is long (V > 0.60 of double-fracture, V > 0.89 of triple-fracture), the σ θmax near a fracture tip increases less than 5%, which is similar to a single fracture, and the interaction of fractures is not significant.

Coplanar fractures interaction
The maximum circumferential tensile stress σ θmax near a tip of coplanar fractures is calculated and the change in stress compared to a single fracture is analyzed at 13:00 with a length of fracture of 0.35 m and a length of fracture opening of 0.14 m (Fig. 12).Horizontal spacing b of double-fracture is 0.01, 0.06, 0.11, 0.16, 0.21, 0.26, 0.31, 0.36 m, as well as horizontal spacing b of triple-fracture is 0.06 m and vertical spacing d is 0.02, 0.04, 0.06, 0.08, 0.10, 0.12 and 0.14 m.The σ θmax near a tip of coplanar double-fracture does not change significantly as horizontal spacing b increases.Fracture interaction is weak and the value of σ θmax is similar to a single fracture.
When the distance between coplanar triple-fracture is small (V ≤ 0.20), the σ θmax near a fracture tip increases more than 20% and the interaction of fractures produces a significant increase in σ θmax .When there is a certain distance between coplanar triple-fractures (0.20 < V ≤ 0.71) , the fracture interaction is weak.When the distance is long (V > 0.71) , the σ θmax near a fracture tip increases less than 5%, which is similar to a single fracture, and the interaction of fractures is not significant.Stress field of a fracture tip is superposed due to fracture interaction.When the spacing between parallel or coplanar fractures is small, the superposition of thermal stress of a fracture tip is significant.As the spacing increases, the value of σ θmax of multiple fractures tends to be the same as a single fracture.

Thermal stress field of fractured rocks
The thermal stress field of fractured rocks in a high-temperature environment is analyzed by numerical simulation method.The boundary for heat transfer between a rock and the external environment is complex and variable as the heat exchange process is affected by several natural factors.Furthermore, an engineering rock mass is characterized by nonuniformity and discontinuity due to intensive development of joints and fractures.Therefore, analytical methods are restricted, and numerical simulation methods are effective technical solutions.

Finite element analysis and related parameters
The finite element model of a fractured rock in the high-temperature environment is 10 m × 10 m × 10 m in size.The triangular fracture develops on upper boundary along z direction with a fracture length of 0.35 m and a fracture opening length of 0.14 m.Since the model size is much larger than that of the fracture, the model is approximately satisfied with a semi-infinite body with the fracture.Parameters of an environment and rock properties are shown in Fig. 4 and Table 1.In the finite element analysis, temperature field and thermal stress field satisfy the Fourier heat conduction law and the equilibrium differential equation, respectively.Simulation of temperature field and thermal stress field is performed using solid heat transfer module, surface-to-surface radiation module and solid mechanics module.
Temperature field of a fractured rock and a non-fractured rock in a high-temperature environment at 13:00 is obtained, as shown in Fig. 13.It is observed that the temperature field of a fractured rock is nonlinear.Temperature around the fracture increases and isotherms around the fracture tip are concave.Temperature under the fracture tip decreases sharply as depth increases, approaching initial rock temperature.The consistent result of numerical simulation and theoretical analysis indicates that the model is rational.However, as a finite body is

Temperature field of fractured rocks
Temperature field of a rock with parallel fractures and coplanar fractures at 13:00 in a high-temperature environment is presented in Figs.www.nature.com/scientificreports/fractures b is 0.01, 0.06 m, respectively.The upper surface of the numerical model is selected for analysis.The results indicate that the temperature around fractures increases significantly.As distance between parallel fractures and coplanar fractures is small, the interaction of multi-fracture is excellent and the regional temperature field between fractures is significantly enhanced.In the same environment, thermal deterioration is more likely to occur in a multi-fractured rock due to high temperature and large temperature range.

Thermal stress field of fractured rocks
Thermal stress field of a rock with parallel fractures and coplanar fractures at 13:00 in a high-temperature environment is shown in Figs.16 and 17.Spacing of parallel fractures d is 0.05, 0.10 m and spacing of coplanar fractures b is 0.01, 0.06 m, respectively.An analysis of upper surface of the numerical model suggests a significant increase in thermal stress around the fractures.The interaction of multi-fracture is great and the regional stress field between fractures is significantly enhanced as the distance between parallel fractures and coplanar fractures is small.In the same environment, thermal deterioration is more likely to occur in a multi-fractured rock.

Case study of rock thermal stress in a tunnel portal
In southern China, a tunnel portal has been exposed to strong solar radiation in a high temperature environment for a long time.Thermal deterioration of fractured rocks occurs as a result of temperature fluctuations and thermal stress concentration.In order to study the changes of fractured rocks thermal stress in a tunnel portal under long-term of solar radiation, the model of this paper is used to analyze the thermal stress of fractured rocks after 5 years and 10 years by numerical calculation.
It is assumed that the tunnel has a circular cross-section, and there is a triangular fracture along z direction with a fracture length of 0.35 m and a fracture opening length of 0.14 m at 1 m above the tunnel entrance, as shown in Fig. 18.The granite property parameters of the tunnel portal are shown in Table 1, and the annual changes of temperature and solar radiation are shown in Fig. 19.The initial condition of finite element model is the temperature of rocks at the initial moment (initial rock temperature).The boundary condition is the heat exchange between the upper surface of the model and the external environment (Eq.( 3)), and the remaining surface is the adiabatic boundary.The boundary of the fracture is the radiation energy absorbed by the fracture  (Eq.( 8)).The lower boundary of the model is a fixed constraint, and the remaining boundaries can expand and shrink freely.
The thermal stress of fractured rocks of a tunnel portal after 5 years and 10 years under solar radiation is shown in Fig. 20.The cross section containing the fracture is selected (Fig. 18) for analysis: (1) The thermal stress around the fracture increases significantly, and stress concentration occurs at the fracture tip.Under long-term solar radiation, the thermal stress at the fracture tip increases to 12.3 MPa after 10 years.(2) The thermal stress field around the fracture is nonlinear, and the range of stress redistribution at fracture tip tends to be wide.If a fracture is close to the tunnel portal, it is easy to expand and even penetrate the rock surface, which is a potential hazard source.The results are consistent with the literature [46][47][48][49][50] .
For exposed rocks of tunnel portals in high temperature environments, various measures can be conducted to reduce thermal stress and prevent thermal deterioration, e.g.shielding measures, grouting and anchor stock, which could reduce the direct solar radiation to rocks or enhance rock mechanical properties.

Conclusions
In this paper, a function expression of triangular heat sources of a fractured rock in a high-temperature environment is established.By the function expression, the nonlinear temperature field and thermal stress field of rock fracture tips are calculated.Additionally, influence of fracture shape parameters and fracture interaction is analyzed.The following conclusions are obtained: 1.During insolation period, the internal temperature of a fractured rock is higher than that of a non-fractured rock.The change in rock temperature field caused by a fracture is concentrated in areas around the fracture.Temperature field near a fracture tip is nonlinear.Temperature of a triangular fractured rock increases as the fracture apex angle increases.2. During insolation period, the thermal stress field of a rock is nonlinear and the stress of fracture tips increases due to fractures.The σ θmax of a triangular fractured rock increases as the fracture apex angle increases.3.As a result of fracture interaction, the thermal stress fields of fracture tips are superposed.When spacing between parallel fractures or coplanar fractures is small, the superposition effect of thermal stress field is evident.In parallel double-fracture, parallel triple-fracture, and coplanar triple-fracture, where the ratio of a vertical spacing to a fracture length is less than 0.20, 0.37 and 0.20, respectively, the σ θmax near a fracture  www.nature.com/scientificreports/tip increases more than 20% and the interaction of fractures contributes to a significant increase in σ θmax .
As the spacing increases, the value of σ θmax approaches that of a single fracture.The interaction of coplanar double-fracture is weak and the change in value of σ θmax is not significant.4. Temperature field and thermal stress field of a fractured rock are both nonlinear in a high-temperature environment.Temperature and thermal stress around fractures increase significantly.The regional temperature field and thermal stress field between fractures are enhanced by multi-fracture interaction.As a result, thermal deterioration is more likely to occur in a multi-fractured rock in the same environment.5.The thermal stress of fractured rocks in a tunnel portal under the long-term of solar radiation is analyzed in southern China.After 10 years, the thermal stress of the rock fracture tip increases to 12.3 MPa, and the stress redistribution range at fracture tip tends to be wide, which is easy to lead fracture expansion and penetration, forming a potential hazard source. Vol

Figure 1 .
Figure 1.Thermal cracking of exposed rocks in high-temperature environments.

Figure 5 .
Figure 5. Calculate points of thermal stress.

Figure 6 .
Figure 6.Temperature-time curves of a rock: (a) a fractured rock; (b) a non-fractured rock.

Figure 7 .
Figure 7. Analysis of temperature change of typical triangular fractured rocks.

Figure 8 .
Figure 8. Thermal stress-time curves of a rock: (a) σ x of a fractured rock; (b) σ y of a fractured rock; (c) σ z of a fractured rock; (d) σ x and σ y of a non-fractured rock.

Figure 9 .
Figure 9. Influence of relative critical size α on maximum circle tensile stress σ θmax .

Figure 11 .
Figure 11.Interaction of stress field of parallel fractures.

Figure 13 .
Figure 13.Temperature field of a rock in a high-temperature environment: (a) a fractured rock; (b) a nonfractured rock (°C).

Figure 18 .
Figure 18.Model of a tunnel portal.
−1 )Thermal stress field caused by a triangular a fracture line heat source could be expressed as: